Method and system for aligning and classifying images

ABSTRACT

In one embodiment, L dimensional images are trained, mapped, and aligned to an M dimensional topology to obtain azimuthal angles. The aligned L dimensional images are then trained and mapped to an N dimensional topology to obtain 2N vertex classifications. The azimuthal angles and the 2N vertex classifications are used to map L dimensional images into 0 dimensional images.

RELATED APPLICATION DATA

This application is a continuation of co-pending U.S. patent applicationSer. No. 15/785,380, filed Oct. 16, 2017, entitled METHOD AND SYSTEM FORALIGNING AND CLASSIFYING IMAGES, which is a continuation of co-pendingU.S. patent application Ser. No. 14/746,836, filed Jun. 22, 2015,entitled METHOD AND SYSTEM FOR ALIGNING AND CLASSIFYING IMAGES, now U.S.Pat. No. 9,792,518, issued Oct. 17, 2017, which is a continuation ofco-pending U.S. patent application Ser. No. 13/960,231, filed Aug. 6,2013, entitled METHOD AND SYSTEM FOR ALIGNING AND CLASSIFYING IMAGES,now U.S. Pat. No. 9,064,188, issued Jun. 23, 2015, which is acontinuation of co-pending U.S. patent application Ser. No. 11/579,481,filed Sep. 16, 2008, entitled METHOD AND SYSTEM FOR ALIGNING ANDCLASSIFYING IMAGES, now U.S. Pat. No. 8,503,825, issued Aug. 6, 2013,which is a PCT National Phase Application of PCT Application Serial No.PCT/US05/15872, filed May 6, 2005, entitled METHOD AND SYSTEM FORALIGNING AND CLASSIFYING IMAGES, which claims the benefit of U.S.Provisional Application Ser. No. 60/569,361, filed May 6, 2004, entitledREFERENCE-FREE TECHNIQUE FOR ALIGNING AND CLASSIFYING EMSINGLE-PARTICLES IMAGED BY RANDOM CONICAL TILT: USE OF N-DIMENSIONALTOPOLOGICAL ARRAYS OF ARTIFICIAL NEURONS, the entire disclosure of eachwhich applications is herein expressly incorporated by reference.

GOVERNMENT SUPPORT

This invention was made with government support of Grant No. EY04 1 10,awarded by the NIH. The government may have certain rights in thisinvention.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates generally to alignment and classificationprocedures and more specifically to alignment and classificationprocedures without a dependence on reference elements.

Background Information

The 3D (3 dimensional) structures of macromolecules can be investigatedby single particle analysis, a powerful technique of electron microscopythat provides more penetrating structural analyses of macromoleculeswhose x-ray crystallography is problematical. Resolution and analyses ofmolecular interactions and conformational changes, using single particleanalysis, have been advancing in pace with the image processing methodsemployed (Ueno and Sato, 2001). Such methods are needed because imagedmacromolecules are obscured by noise backgrounds (with SIN ratios [SNR]often below unity). Noise has its origin partly in the low irradiationlevels used to minimize specimen damage. To attenuate noise, one mustaverage the 2D images with the same 3D orientations, a process requiringoptimal alignments among the gallery of data set images. Signals thenreinforce one another, and noise tends to average out.

As with other techniques of 3D reconstruction, 2D image alignment iscritical when applying the RCT (random conical tilt) method to fixedparticles, imaged either “face up” or “face down” in a membrane bilayeror film. To recover 3D information coherently, one must know therotational orientation of each data set image in the plane of thebilayer. This is normally accomplished by bringing untilted images{those viewed “head on” at 0°) into a common rotational alignment. Oncerelative rotational orientations (the amounts of rotation of givenimages required to bring them into common alignment) are known foruntilted images, they also become known for tilted images, because ofpairwise imaging. Single particle analysis should, in principle, achieveatomic resolution. In practice, however, various circumstances preventthis.

Normally, one assumes the existence of a prototype reference image,against which the data set images can be aligned. However, thisassumption is unjustified for inhomogeneous data sets. Another problemis that alignments are biased by the choice of reference images. Onemethod to reduce this bias is to average the images aligned to aparticular reference, to yield a revised reference (Penczek et al.,1992). However, when the images have a poor SNR, or represent differentviews of the same macromolecule, this procedure yields a final averagedreference that shares features of the original reference. “Iterativereference free alignment,” selects and aligns two images at random. Theprocess is then repeated with a third selected image, etc., until thedata set images have been exhausted (Penczek et al., 1992). However,because the order of selection biases results, the process is repeatedwith random orders of selection, thereby reducing the bias because thismethod uses a changing global average for reference, it is not strictlyreference free (van Heel et al., 2000).

The “state of the art” technique for generating and aligning referenceimages representing different views of a data set, is designated“Multivariable Statistical Analysis/Multi Reference Alignment” (MSA/MRA;see Zampighi et al., 2004). Some variations between data set images maynot reflect structural differences, but merely positional differences(e.g., in plane rotational orientation, in the case of RCT). For thatreason, it is undesirable, when classifying data sets, to consider inplane rotational differences to be valid. Consequently, beforeclassification, images must be brought into mutual rotational alignment,thereby eliminating particle orientation as an independentclassification variable.

However, alignments using correlation based techniques are only welldefined operations when galleries of images are homogeneous. But toproduce representative classes, data set images must first be aligned,which requires an initial set of representative classes. To cope withthis “circularity,” workers have resorted to iterative cycles ofclassification and alignment using MSA/MRA, until results stabilize.However, this procedure does not guarantee attainment of the globalminimum of the “energy” function. In addition to these shortcomings,MSA/MRA hinges on subjective operator choices of many critical freevariables that impact the final result. Consequently, such resultstypically are operator dependent. Finally, MSA/MRA often consumes monthsof processing (Bonetta, 2005).

In one embodiment, the technique developed here aligns, or orients, dataset images produced by RCT by directly classifying their relative inplane rotational orientations, without rotating images into “alignment”with references, common to the above described techniques. Instead ofstarting with selected reference image(s), one typically starts withover 8 million random pixel values. Coupling this procedure with asufficiently small influence of the data during each training cycle,eliminates training bias because the alignment results becomeindependent of the order of the images used to train the network.

This alignment procedure bypasses the need for alternate cycles ofalignment and classification in order to achieve alignment. Itsimultaneously classifies both structural differences and rotationalorientations of images during each training cycle. It is a selforganizing process, performed on the surface of a cylindrical array ofartificial neurons. After reference free alignment, a more detailedclassification according to structural differences may be required.Zampighi et al. (2004) found that vertices of the square planar SOMs'(self organizing maps') map differing views of RCT imaged particlesembedded in a membrane bilayer. But the homogeneity of vertex partitionsgenerated by a 2-CUBE SOM is imperfect when the SNRs are low, or thedata sets are heterogeneous. Also, the number of vertex classes islimited to four. To obviate the heterogeneity arising from 2D SOMs, andlesser restrictions on the number of vertex classes, we developed amethod called “N Dimensional vertex partitioning” with which a dataset's principal classes “migrate” to the vertices of ND (N dimensional)hypercubical arrays. Because an ND hypercube has 2^(N) vertices, itpartitions 2^(N) vertex classes.

We found that, as the dimension is stepped up (at least to a value ofN=4), the average homogeneity of the vertex classes improves. Thislikely eventuates from the high degree of data compression that occursin mapping high D data onto low D grids. The higher the grid dimension,the less the data compression, and the greater the resultinghomogeneity. If the operator desires two vertex classes of the data set,a value of N=1 is selected, for four vertex classes, a value of N=2, foreight vertex classes, a value of N=3, etc. This allows one to controlthe number and homogeneity of the generated vertex classes.

In one embodiment, we call the combination of these reference freemethods, SORFA, for “Self Organizing, Reference Free Alignment.” SORFAbenefits from the intrinsically parallel architecture andtopology/metric conserving characteristics of self-organizing maps. Herewe demonstrate SORFA using the Kohonen self-organizing artificial neuralnetwork, an unsupervised method of competitive learning that employs notarget values decided upon in advance (Deco and Obradovic, 1996). Theinspiration and methodology for this type of network are based on thecircumstance that most neural networks comprising the mammalian brainexist in 2D arrays of signal processing units. The network's response toan input signal is focused mainly near the maximally excited neuron.

In one embodiment, SORFA uses this methodology to classify imagesaccording to azimuthal orientations and structural differences. Itbenefits from the SOMs' insensitivity to random initial conditions.Further, for alignment there are only two critical free parameters:learning rate, and circumference/height ratio of the cylinder. SORFA'sspeed benefits from its intrinsically parallel architecture. A data setof 3,361 noisy, heterogeneous electron microscopical {EM) images ofAquaporin 0 (AQP0), for example, was better aligned in less than 9 min,than was a MSA/MRA run requiring six months. SORFA was far simpler touse, and involves far fewer chances for operator error, as was suspectedin the above six month MSA/MRA alignment. These AQP0 channels were fromthe plasma membranes of native calf lens fibers, which are tetramers, 64Å on a side, MW˜118kDs (Konig et al., 1997).

Images of the AQP0 channel obtained by RCT geometry already have beenaligned and classified using the alignment through classificationapproach (Zampighi et al., 2003). By using knowledge gained in applyingSORFA to test images, we aligned and classified the AQP0 images.Finally, we compared the structure of the 3D reconstructions to theatomic model of Aquaporin I (AQP I), a closely related and structurallysimilar channel with 43.6% identity and 62.6% similarity to AQP0 (Gonenet al., 2004).

BRIEF DESCRIPTION OF THE FIGURES

FIGS. 1A-1F illustrate flowcharts illustrating a reference-free methodfor aligning and classifying images, according to one embodiment of theinvention.

FIG. 1G illustrates the effect of noise on vertex-class contamination,according to one embodiment of the invention.

FIG. 2 illustrates a schematic diagram of a reference-free method foraligning and classifying single-particles, according to one embodimentof the invention.

FIG. 3 illustrates azimuthal angle classification and alignment ofsynthetic population of white squares, according to one embodiment ofthe invention.

FIG. 4 illustrates azimuthal angle classification and alignment of theAQPI test set, according to one embodiment of the invention.

FIG. 5 illustrates the effect of mapping variable numbers of testclasses for N=2, according to one embodiment of the invention.

FIG. 6 illustrates azimuthal angle classification and alignment of AQP0,according to one embodiment of the invention.

FIG. 7 illustrates AQP0 2 dimensional vertex classes, according to oneembodiment of the invention.

FIG. 8 illustrates a comparison of an AQP0 population A member with theextracellular domain of the AQPI x-ray model, according to oneembodiment of the invention.

FIG. 9 illustrates an AQP0 Population B representative compared to AQPIx-ray model cytoplasmic domain, according to one embodiment of theinvention.

FIG. 10 illustrates an orientation of amino termini in a population-Bmold, and the polypeptide loops in a population-A mold, according to oneembodiment of the invention.

FIG. 11 illustrates an analysis of footprints, according to oneembodiment of the invention.

FIG. 12 illustrates the angular deviation from linearity as a functionof the number of iterations, according to one embodiment of theinvention.

FIG. 13 illustrates an unsymmetrized azimuthal angle classification andalignment of AQP0, according to one embodiment of the invention.

FIG. 14 illustrates replicas generated from symmetrized andunsymmetrized alignment and classification, according to one embodimentof the invention.

FIG. 15 illustrates a comparison of test runs using non-aligned, SORFAaligned, and MSA/MRA aligned AQP0 images, according to one embodiment ofthe invention.

FIG. 16 illustrates alignment and hyperalignment, according to oneembodiment of the invention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

1. Overview Figures

FIGS. 1A-1F illustrate a method of aligning and classifying images,according to one embodiment of the invention. In one embodiment, areference-free procedure is performed to circumvent a dependence onreference elements common to other correlation-based alignmentprocedures. FIGS. 1A-1C illustrates a general process, and FIGS. 1D-1Fillustrate a specific example, using AQP0 images.

FIG. 1B sets forth details of step 101 of FIG. 1A, which trains and mapsto an M dimensional topology using L dimensional images to obtainazimuthal angles. The input images are also brought into alignment byapplying the calculated azimuthal angles in order to test the accuracyof the alignment. In step 105, images from a data set comprising Ldimensional images are input. In step 106, metric starting values basedon the M dimensional topology are input. The M dimensional topology isbased on the topology of the L dimensional images (to obtain a closematch). In step 107, the data set symmetry and the azimuthal angles aredetermined based on the input information. In step 108, the input Ldimensional images are rotationally aligned by applying the azimuthalangles to the input L dimensional images. In step 109, steps 106-107 arerepeated, using the aligned images as the new input images, to determineif the azimuthal angles are correct.

FIG. 1C sets forth the details of step 102 of FIG. 1A, which trains andmaps the N dimensional topology using the aligned L dimensional imagesto obtain vertex and/or neighborhood classifications. In step 110, thedimensions (the metrics) of the N dimensional topology, training cycles,initial learning rate, and the optional initial learning radius areinput. In step 111, the vertex and/or neighborhood classifications aredetermined based on the input information.

As explained above, FIGS. 1D-1F illustrate an example of the method setforth in FIGS. 1A-1C, according to one embodiment of the invention. Thedata set Aquaporin 0 (AQP0) is used in this example. It should bestressed that this data set is only an example, and many other types ofdata sets may be used.

Referring to FIG. 1D, in step 112, 2 dimensional untilted images (notethat L=2 in this example) are used to train and map a cylindricaltopology (note that M=3 in this example) to obtain azimuthal angles. Theazimuthal angles are then applied to the untilted images to producealigned images so that the alignment can be tested and step 113 can beperformed. (Note that the AQP0 data set is a random conical tilt dataset, which uses untilted and tilted images.) The aligned 2 dimensionaluntilted images are then trained and mapped to an N dimensional topologyusing the aligned images to obtain 2^(N) vertex classifications in step113. In step 114, the azimuthal angles and the 2^(N) vertexclassifications are used to map 2 dimensional images into 3 dimensionalimages (3D reconstructions). In step 115, step 112 (the training,mapping, and aligning step) is repeated using one or more vertex classsubsets obtained from step 113, in order to hyperalign the 2 dimensionaldata. Hyperalignment may be repeated as necessary by repeating step 113followed by step 112, using one or more vertex classes of 2 dimensionalimages.

FIG. 1E sets forth details of step 112 of FIG. 1D, which trains and mapsto a cylindrical topology using the L dimensional images to obtainazimuthal angles. The L-dimensional images are then brought intoalignment by applying the azimuthal angles. In step 116, 3361 2dimensional images from the AQP0 data set are input. In step 117, fivestarting values, based on the cylindrical topology are input:circumference to height ratio, a specific circumference based on thedesired resolution of the azimuthal angles, the number of trainingcycles, an initial learning rate, and learning radius.

Note that in this step, SORFA (Self-Organizing Reference-Free Alignment)starts with a cylindrical topology, one that is naturally suited toclassifying azimuthal angles of images obtained from random conicaltilt. One starting value is known (the output grid resolution=thecylinder circumference or angular resolution desired), but two startingvalues must be adjusted (the initial learning rate, and the cylindercircumference to height ratio) in order to limit the error to anacceptable level as measured by the test run. The other parameters(cylinder circumference, number of training cycles and initial learningradius) are well-defined.

Note that SORFA sometimes requires lower initial learning rates withdata sets of greater heterogeneity. For tests with EM images, forexample, learning rates may be set to values of 0.001 or lower. A lowlearning rate makes it so the first few training images don't bias thenetwork. For a discussion of the effect of the initial learning rate,see Example 2.

In step 118, the symmetry of the 2 dimensional images as well as theidentity of the azimuthal angles are determined based on the inputinformation. SORFA outputs a list of relative azimuthal angles (referredto as Output B) for all the untilted 2 dimensional images of inputimages along one of two mutually perpendicular axes on the surface ofthe cylinder of artificial neurons (the circumferential axis).

SORFA also outputs a determination of symmetry (referred to as output A)in order to properly calibrate the azimuthal axis.

In step 119, the input 2 dimensional images are rotationally aligned byapplying the azimuthal angles to the input 2 dimensional images, by wayof an anti-aliasing rotator.

In step 120, steps 117-118 are repeated to determine (test) if theazimuthal angles are correct. These steps are repeated using the alignedimages, and the same starting values.

The test run is the reprocessing of aligned images using the samecylinder conditions, in order to determine angular consistency betweenruns. In the test run, the greater the deviation of the mapped imageangles from the mean, the greater the relative angle deviation betweencorresponding images of the two runs. Therefore, one seeks the narrowestvertical band of images, stretching from top to bottom of the testcylinder. This may be expressed quantitatively as the standard deviationof the angle distribution in output B of the test run.

The new output B gives the standard deviation of the distribution ofazimuths. It is than determined if the standard deviator is within anacceptable range. If no, the process moves back to step 116 and repeats,but with a modified circumference to height ratio and learning rate instep 117. If yes, the process moves to step 121.

FIG. 1F sets forth the details of step 113 of FIG. 1D, which uses theuntilted 2 dimensional images to train and map to an N dimensionaltopology to obtain 2^(N) vertex classifications. In step 121, the valueN (for example a value of N=3 will return 8 vertex classes), the edgelength of the N dimensional hybercube, training cycles, and an initiallearning rate and learning radius are input.

The dimension of the hypercube (or hyper-parallelepiped) is set to N forthe network to return untilted images in 2^(N) vertex classes from thedata set (representing 2^(N) different views of the imaged macromolecule).

In step 122, the 2^(N) vertex classifications are determined based onthe input information. For a discussion on locating the vertices of thehypercube see Example 1. If only two classes are desired, they may begleaned directly from the cylinder. No additional classification isnecessary.

Embodiments of the invention have been described above, which provide amethod for aligning and classifying images, comprising training,mapping, and aligning L dimensional images to an M dimensionaltopological array of L dimensional neuron images to obtain azimuthalangles and/or additional Euler angles; training and mapping the alignedL dimensional images to an N dimensional topological array of Ldimensional neuron images with vertices to obtain vertex and/orneighborhood classifications; and mapping the L dimensional images to 0dimensional images. In one embodiment, the initial learning rate is setto be low so that the first few images of the data set do not bias thenetwork. In another embodiment, the alignment is based on a randomrotation of input images and is thus independent of any input image'sorientation, and linearly calibrates the azimuthal axis. In anadditional embodiment, the circumference to height ratio is picked basedon a small circumference, and the circumference is scaled up, keepingthe circumference to height ratio constant, so that higher resolutionazimuthal angles are obtained. In another embodiment, the trainingcomprises morphing the L dimensional neuron images so that they areprogressively greater in similarity to the input L dimensional images sothat progressively smaller neighborhoods of the L dimensional neuronimages, as measured in the topological space after morphing, tend to beprogressively greater in similarity. In a further embodiment, themapping is done based on the similarity between the L dimensional neuronimages and the L dimensional images. In yet another embodiment, randompixel values are used for the L dimensional neuron images that morphtoward the input L dimensional images. (Note that random values do notnecessarily need to be perfectly random, just somewhat disordered.) Inanother embodiment, symmetry further comprises reviewing the mapped Ldimensional images to determine how they much and in what direction themapped L dimensional images rotate around the circumference of thecylinder, the cylinder comprising of the L dimensional neuron images. Inan additional embodiment, the metrics and topologies between the Ldimensional images and the M dimensional array of L dimensional neuronimages are matched in order to establish meaningful coordinate systemsin such spaces, which convey the angular and/or class information of theL dimensional images, based on where they map in their respectivecoordinate systems. In a further embodiment, determining if theazimuthal angles and/or the additional Euler angles are correctcomprises checking to see if the azimuthal angles and/or the additionalEuler angles are tightly constrained. The invention is not limited tothese embodiments, and those experience in the art will see thatmultiple additional embodiments are possible.

2 Additional Information on Materials and Methods

2.1 AQPO Data Sets

AQP0 belongs to one often mammalian families of highly conserved, waterselective, membrane channel proteins. Each monomer of aquaporins isthought to be wedged tightly within the tetramers, and to possess anindependent water pore, 6 tilted (18 30°), membrane spanning a helicesthat form a barrel, and 5 polypeptide loops (A-E) that connect thehelices. Two of these loops (B=cytoplasmic, and E=extracellular) areconserved throughout the aquaporin superfamily. Each contains theaquaporin signature motif, namely, asparagine proline alanine (NPA).Loops B and E are partially helically coiled and fold into the bilayerfrom opposite sides to form key elements of the pore wall. Theseinfolding loops are thought to be essential for maintaining theconnectivity of water flow in the narrow pore channel.

Most such channels have been studied by x-ray and electroncrystallography, but AQP0 has not yet been crystallized. Each monomercontains a curvilinear pore pathway, consisting of three domains:external and cytoplasmic portions that flare out into vestibules about15 Å wide; and a constricted region lying between them, the latterconsists of a narrow, 20 Å long selectivity filter, only 2.8-3.0 Å indiameter at its narrowest point, which is only one amino acid residuelong. The unusual combination of a long, narrow channel lined withhydrophobic residues, with only a few solute binding sites, facilitatesrapid water permeation, with a minimal energy barrier, but preventstransfer of most other small molecules and ions. These unusual featuresowe largely to a rich potential for interactions between differentdomains and subdomains of the continuous polypeptide chains—eachcontaining a helices, connecting loops, hundreds of side chain residues,and amino and carboxylic termini that extend from the cytoplasmic side(Cheng et al., 1997; Murata et al, 2000; Sui et al., 2001; NemethCahalan et al., 2004).

We used a data set of 3361 AQP 0 (AQP0) images generated as described inZampighi et al. (2003). Purified AQPO was reconstituted in liposomes,which were freeze fractured, multi axis shadowed with platinum andcarbon, and imaged at 0° and 50° using RCT (Radermacher, et al., 1987).These freezing, fracturing, and shadowing techniques lead to highretention of native conformation (Morris et al., 1994). By attachinglarge liposomes (˜2 μm diameter) to glass before fracturing, the localcurvature of the lipid bilayers was flattened, thereby effectivelyeliminating two of the three Euler (azimuthal) angles as free variables.An innovative pulse-shadowing protocol significantly increased replicastability (Zampighi et al., 2003) and, may have limited platinum graingrowth.

By employing this protocol, consisting of two 4-sec exposures, spaced2-3 sec apart, the platinum grain diameter may have been reduced. The2-3-sec “cooling” interpulse also may have led to reduced kineticenergy/mobility of surficial platinum atoms. This would lead to smallergrain sizes, kinetics of dewetting being constrained by atomic mobility(Woodward and Zasadzinski, 1996). Apparent smaller grain size, anddeeper penetration, on pulsing, also increase replica mechanicalstability (Zampighi et al., 2003). Inasmuch as migration of shadowedplatinum atoms is precluded (Woodward and Zasadzinski, 1996), it isevident from the replicas obtained (FIGS. 8-11, 14), particularly viewsof their convex and concave sides (FIG. 14), that extensive platinumpenetration of the samples occurred.

2.2 Pre-Processing of AQP0 Data Sets

Low pass filtering, density inversion, application of a circular mask,and translational alignment, for both tilted and untilted images, aswell as imposition of C4 symmetry on the untilted images, were performedwith the Imagic-5 software package (van Heel et al., 1996). Allsubsequent processing of untilted images used the SORFA neuronalnetworks described below. Care must be taken not to mask the images tootightly, otherwise artifacts may result that may interfere with properalignment. Untilted images were filtered to 24 Å for the alignment andclassification stages of SORFA.

2.3 Construction of SORFA

SORFA is a Kohonen SOM [or any other SOM that consists of 1 or higherdimensional arrays of artificial neurons (continuous or discrete),codebook vectors, or tensors, all of which morph toward, or approach acondition of similarity to, the training images, in a way that attemptsto preserve the topology of the data set, like, for example, fuzzyKohonen clustering networks (FKCN) (Pascual et al., 2000), KerDenSOMs(Pascual Montano et al., 2001), and Parameterized Self-organizing Maps(PSOM), where continuous mappings may be attained (Walter and Ritter,1996) that can switch between any number of different topologies andmetrics in a serial and/or parallel fashion, with the results of one SOMbeing the input for the next. The application of cylindrical and Ndimensional hypercubical topologies, in particular, will be the focus ofthis paper, because they match the topological characteristics of thedata used here. A mathematical account of the operation of the KohonenSOM may be found in Zampighi et al. (2004). In constructing thecylindrical topology, we wrapped the planar grid SOM around itself andseamlessly attached opposite edges to form a cylindrical array ofartificial neurons (see FIG. 2). When using the cylindrical topology,training proceeds by singly presenting randomly rotated images (theuntilted 2D EMs), in a random order, from the data set to the network,and allowing gradual adaptation of the grid images. Letter R in FIG. 2symbolizes this operation. Rotation is performed by an anti-aliasingrotator.

Because translational alignment of the untilted images has already beencarried out, all rotation is done around image centers. Results aredisplayed on the cylindrical grid, after “training” the network, bymapping the original unrotated images making up the data set onto thesurface of the cylindrical grid (letter M in FIG. 2). Azimuths of thedata set images then are read directly from the azimuthal axis of theoutput cylinder. Distances are calculated only along the cylinder'ssurface (for details see Example 1). After the azimuths of the data setimages are read from the azimuthal axis of the output cylinder, they areapplied to the images, thereby producing the required rotationalalignment. The topology of the network is then switched to an ND cubicalarray, to allow operator control on the number of vertex classesgenerated, and the degree of vertex class homogeneity. No randomrotation of the images is needed during training at this step. Examplesof ND arrays for values of N from 0 to 4 are shown in FIG. 2. For thesearrays, simple Euclidean distances are used when calculating distancesin neuron space. These features were implemented in XMIPP (Marabini, etal., 1996) and in the SOMULATOR, an artificial neural network that wedeveloped to run on Microsoft Windows based platforms.

2.4 Operation of SORFA

2.4.1 Azimuthal Angle Classification

The following steps are used to determine the relative azimuthal anglesof the untilted input images, as schematically represented in FIG. 2.SORFA starts with a cylindrical topology, one that is naturally suitedto classifying azimuthal angles when the other two Euler angles aretightly constrained:

The operator switches the topology of SORFA to cylindrical, and inputs 3starting values: one known (output grid resolution=cylindercircumference), and two that must be adjusted (learning rate, andcylinder circumference/height ratio), in order to limit the error (avalue obtained from the test run in step 6) to a level acceptable to theoperator. The number of training cycles for the final run is usuallyfixed at ˜300,000, [Note: if the sampling rate of the camera exceeds,1/3.1 Å, then larger numbers of iterations may be needed to achieve thebest results] and the initial learning radius is automatically set tothe circumference of the cylinder, though this requirement is not rigid.A shortcut is to work, first, with smaller grids, until ideal conditionsare found, then to scale up to the desired circumference, keeping thecircumference/height ratio approximately constant.

Note: The cylindrical topology may correspond to a regular or irregularcylinder or hypercylinder, or any other volume or hypervolume with acylindrical topology. All the demonstrations in this paper employ aregular cylinder. An irregular cylinder is a frustum of the surface of acone. Such a frustum may be formed from the surface of a cone with acircular base by cutting off the tip of the cone, forming a lower baseand an upper base. In the case that the upper and lower bases arecircular and parallel, and mutually perpendicular to the height, thenthree variables uniquely specify this irregular cylinder: thecircumference of the two bases, and the distance separating them.

The operator inputs the 2D images (or any ND entities representable asvectors or tensors) into SORFA and starts training the network. Runstypically last a few minutes to a few hours on a 1.33 GHZ AMD/MP 1500+dual processor (processing time proportional to number of trainingcycles entered.

In order to properly calibrate the azimuthal axis, overall symmetry ofthe data set must be determined. Inspection of output “A” (output A)reveals by how many degrees the grid images rotate through onerevolution around the cylinder dividing this value into 360° yields thesymmetry fold. One may use the XMIPP or the SOMULATOR display window toscroll the images across the screen around the direction of thecylinder's circumference (the horizontal direction): this procedureplays a movie of the images around the circumference, working very wellwith noisy data sets to determine the direction and amount of theirrotation.

Output “B” (output B) displays a list of relative azimuths of inputimages along one of two mutually perpendicular axes on the surface ofthe cylinder of artificial neurons (the circumferential axis) by notingthe azimuthal axis coordinate of each image.

2.4.2 Test Run

Input images are brought into rotational alignment by applying theazimuthal angle values displayed in output B to the data set images, byway of an anti-aliasing rotator. The above steps are repeated using thealigned images as the training set, and the same starting values(learning rate and cylinder circumference/height ratio).

The new output B gives the standard deviation of the distribution ofazimuths. If this deviation is within an acceptable range for thedesired angular resolution, one proceeds. Otherwise one returns to thebeginning, but with modified starting values.

2.4.3 Structural Difference Classification

To obtain 2^(N) vertex classes of the data set (representing 2^(N)different conformations, functional states, or views of the imagedmacromolecule), the operator switches over to an ND hypercubicaltopology, or a multidimensional rectangular grid topology. Thesetopologies appear to be naturally suited for classifying structuraldifferences [Note: at this point, to generate classes, it is possible touse other classification schemes, such as MSA, if one prefers]. Thenetwork is trained with the aligned images for a number of cycles equalto from ˜(100-1000) times the number of images in the data set, at aninitial learning rate of 0.1.

˜(150 -200) images are harvested from each of the 2^(N) vertex classes,also called the principal classes. Non-principal classes may also beharvested from one or more neighborhoods in the ND hypercubical space,each neighborhood representing a homogeneous class. Faster camerasampling rates(>1/3.1 Å), and improvement in other factors required forbetter resolution including, but not limited to, decreased sphericalaberration(<2×106 nm), decreased electron wavelength(<0.00335087 nm),reduced defocus maxima [<(200-300) nm], etc., may require larger numbersof images to be harvested in order to achieve the best resolutionresults. After training SORFA using the ND hypercubical topology andselecting principal and/or non-principal classes, the operator has thechoice to proceed directly to the 3Ds in step 10, and/or refine (orrecalculate) the azimuthal angles by executing step 9 (hyperalignment,see Example 5) before proceeding to the 3Ds.

For a discussion on locating the vertices of the hypercube, seeExample 1. If only two classes are desired, they may be gleaned directlyfrom the cylinder. ˜(150-200) untilted input images that are mapped tothe top most rows of the cylinder are used to create a 3Dreconstruction, representing one conformation of one domain of theimaged protein. ˜(150-200) more images are gathered from the bottom mostrows and are used to make a second 3D reconstruction representinganother conformation of the same or different domain. When gatheringimages, take the images from the top most or bottom most row and thenadd additional rows until ˜(150-200) are obtained.

2.4.4 Hyperalignment (Optional) (Also see Example 5)

The selected principal and/or non-principal classes produced from steps7 8 are each used as a set of input images for independent runs thoroughsteps I to 5 (Note: as a matter of practice, it is better to assemblethe class member images from the original images, because compoundedrotations of a single image, using a rotator with even a small amount ofaliasing error, will introduce progressive errors). Each time step 5 iscompleted, the new output B gives the standard deviation of thedistribution of azimuths. If this deviation is within an acceptablerange for the desired angular resolution, one proceeds to the 3D's instep 10. Otherwise, one repeats this step, but with modified startingvalues (learning rate, and cylinder circumference/height ratio).

2.5 Setting Parameters for Azimuthal Angle Classification

To obtain the best alignment of a data set, only two parameters must bedetermined by trial and error: first, the cylinder's circumference toheight ratio, and second, the initial learning rate. All otherparameters are fairly well defined.

2.5.1 Trial and Error Parameters

Initial learning rate. SORFA normally requires lower learning rates withdata sets of greater heterogeneity. For tests with EM images, forexample, we found that learning rates should be set to values of 0.001or lower. For less heterogeneous, artificially generated, data sets, avalue of 0.01 is sufficient. For a discussion of the effect of theinitial learning rate, see Example 2.

(b) Circumference to height ratio. Empirically, it appears that thegreater the heterogeneity of the data set, the smaller the neededcylinder circumference/height ratio, other variables being heldconstant. Circumference/height ratios often work well over wide ranges.For example, the 3,361 image AQPO data set produced the lowest standarddeviation test runs with circumference/height ratios in a range from0.82 to 1.22 and beyond (for the hyperalignment of ˜(167-182) imagesubsets, we used circumference/height ratio of 20.0). To determine anoptimal circumference/height ratio, it saves time to start with a verysmall circumference (the serial computing time is proportional to thesurface area of the grid). Once the best test run is found for the smallgrid, one can then scale up to the desired larger circumference, keepingthe circumference/height ratio constant, and using the same number ofiterations and learning rate.

2.5.2 Well Defined Parameters

Cylinder circumference. The cylinder circumference is set to whateverangular resolution the operator hopes to achieve (realizing that angularresolution is fundamentally limited by, among other things, noise andthe sampling rate of the EM images). For example, if one wants theazimuthal axis to be calibrated in 6° increments, one uses a cylindercircumference of {360° I [(symmetry)×(6°)]}=360°/(4×6°)=15 neurons.

(b) Initial learning radius (initial Gaussian curve standard deviation).This is set to the largest dimension of the cylinder (usually thecircumference), though a wide range of values typically work.

2.6 Initial Grid Images

It is important that the initial pixel values of the grid images arerandomized. Operators of Kohonen SOMs typically use a shortcut, however;the initial grid images are chosen from the training set, itself, toreduce the number of iterations necessary for output grid imageconvergence. In one embodiment, this shortcut is not be used with SORFAusing a cylindrical topology, because it prevents the global alignmentof heterogeneous populations, by introducing a small rotation of theimages along the height axis, when all rotations should be confined tothe azimuthal axis.

2.7 Random Rotation

Randomly rotating the data set images (Operation “R” in FIG. 2) duringcylinder training serves important functions. First, it minimizes theinfluences of any preferential orientations that might exist in the dataset. Because of the SOM's metric preserving characteristics, suchorientations would bias the network, and introduce non linearity in thecalibration of the cylinder's azimuthal axis. In other words, a nonuniform presentation of angles results in a non uniform azimuthal axiscalibration, because of the SOM's attempt to conserve the metric.Therefore, randomly rotating the images during training is a means oflinearly calibrating rotational variation around the cylinder, that is,a means of linearly calibrating the azimuthal axis in the cylindricalcoordinate system of the output grid. Calibration of the azimuthal axiscan be made arbitrarily linear, given a sufficient number of randomlyrotated presentations of training images (Example 3).

In addition to linearly calibrating the azimuthal axis, random rotationhelps to ensure the very existence of an azimuthal axis. It ensuressufficient angular variation in the training set to guarantee that thenetwork will map azimuths along one of the principal network axes (forthe cylindrical topology, the azimuthal axis wraps around the cylinder).Because the network attempts to preserve the metric of the data, alarger amount of angular variation (more randomly presented images)leads to a guaranteed mapping domain around the cylinder.

2.8 Training Cycles

For final alignment runs, we seek a smooth distribution of orientationsof the presented images during training. This is done in order tolinearly calibrate the azimuthal axis. To achieve this, we recommend anumber of training cycles in the range of ˜100,000 to ˜300,000 [Note:camera sampling rates greater than 1/3.1 Å may require larger numbers oftraining cycles for best results]. Trial and error shows that this rangeworks best (theoretically justified in Example 3). When searching forthe best cylinder height and initial learning rate, one can use a lowernumber of iterations (to save time) and then scale up to 300,000 for thefinal alignment run.

2.9 Evaluating Parameters for Azimuthal Angle Classification

We found that a narrow width {the low standard deviation) of thehistogram of the test run is the litmus test for evaluating azimuthalangular alignment. We generated the histogram by plotting the number ofmappings to a given column of the cylinder versus the column location.Whenever one gets a narrow test run, the azimuths {obtained in output B)yield an alignment that is plainly evident upon visual inspection {see“aligned training set” examples in FIGS. 3, 4, and 6), as well as uponcomparing the azimuths with their known values {when known).

The test run is the reprocessing of aligned images using the samecylinder conditions, in order to determine the consistency of thedetermined azimuthal angles between runs (steps 1 4). In the test run,the greater the deviation of the mapped image angles from the mean(measuring along the circumferential-axis from the cylinder column wheremost of the images have mapped), the greater the relative angledeviation between corresponding images of the two runs. Therefore, oneseeks the narrowest vertical band of images, stretching from top tobottom of the test cylinder. This may be expressed quantitatively as thestandard deviation of the angle distribution in output B of the testrun.

2.10 Setting and Evaluating Parameters for Structural DifferenceClassification

The dimension of the hypercube (or hyper parallelepiped) is set to N forthe network to return 2^(N) vertex classes from the 2^(N) hypergridvertices (FIG. 2). Also, the number of training cycles should be set to˜3 times, or more, the number of training images. We found it ispreferable to use a number of training cycles of ˜100-1000 times thenumber of training images. Better class homogeneity was obtained withhigher values. The learning radius is set to the length of thehypercube. The learning rate may be set to 0.1 initially.

Evaluating the quality of the structural difference classification isnot as straightforward as evaluating the alignment, because, withKohonen SOMs, there is no single variable that measures overall qualityof maps (Erwin et al., 1992; Kohonen, 2001). Two output variables may beconsidered: 1) the quantization error; and 2) the runner up, winningimage distance (Kohonen, 2001). The quantization error gives an overallmeasure of how well grid images in output resemble the training setsmaller values being better. The runner up, winning image distance is ameasure of how well the topology of the data has been preserved on themap. Here too, smaller values are more desirable.

2.11 Preparation of Test Sets Using the AQP 1 X-Ray Model

We took the Aquaporin 1 (AQPI) x-ray model (Sui et al., 2001) and splitit in half. Each half was then reprojected to obtain the extracellularand cytoplasmic domains viewed head on. These two images were used toconstruct a test set for azimuthal classification, by randomly rotatingtheir contents through all angles and adding Gaussian white noise usingImagic 5.

A second test set was created to assess structural differences. The twox-ray AQPI projections (just described) were concatenated to 6 moreimages, which had been generated by imposing different symmetries,ranging from 3 to 6 fold, on the x-ray AQP1 projections. We call these 8test images generator images, because each was then used to generate aclass of test images, as follows: the 8 generator images were modifiedby adding Gaussian white noise and introducing a limited random rotation(5°≤θ≤5°). The SNRs of the test images were calculated, as follows:first, the variance of the signal (before adding noise) was normalizedusing the Imagic 5 Norm Variance utility; noise of a specified variancewas then added using the same utility. The ratio of the variance of thesignal to the variance of the noise is defined to be the SNR (Frank,1996).

2.12 Three Dimensional Reconstruction Using Azimuthal Angles and ClassesObtained from Sorfa

The orientation of a particle in 3 space is completely specified bythree independent variables. For replicas of proteins embedded inphospholipid bilayers imaged using RCT (with a fixed 50° tilt angle) andglass as a flat adhesion surface for the liposomes duringfreeze-fracture, two variables are tightly constrained, and effectivelyeliminated as free variables. Therefore, only one unknown of particleorientation is needed to calculate 3D reconstructions: the relativerotational orientation in the bilayer plane (azimuth). Homogeneousdomains can be determined by partitioning the 2D data into homogenousclasses. If done properly, images representing the cytoplasmic andextracellular domains of the protein should cluster into differentpartitions. The alignment and classification operations were performedby SORFA. After obtaining the Euler angles, tilted images were rotatedto bring the tilt axis into coincidence with the Y axis. The 3Dreconstructions were computed by employing the weighted back projectionalgorithm (Radermacher et al., 1987; Radermacher, 1988), and a Fouriermethod, rooted in the discrete Radon transform (Lanzavecchia et al.,1999).

Euler angles were refined by comparing each tilted image withprojections obtained from the 3D reconstruction. These comparisonsallowed the assignment of new angles to the images, based on the closestmatch with the projections. This refinement, which hinges on theprojection matching strategy of Harauz and Ottensmeyer (1984), wasrepeated several times until convergence. Imprints of the concavesurfaces were calculated as described by Zampighi et al. (2003). 3Dreconstructions were displayed with the AVS package (AdvancedVisualization System, Inc.) at levels of isosurfacing chosen to obtain athickness of the replica (10.5 15.5 Å) consistent with the 13(±2) Åthickness of the platinum-carbon replica (Eskandari et al., 1998, 2000).

3. Results

One of our principal goals was to develop a reference free method ofaligning and classifying images with different orientations in theplane. Below, we describe examples using artificially created data sets,and then AQP0 EM images.

3.1 Tests Using Artificially Generated Images

3.1.1 Azimuthal Angle Classification

We generated a synthetic population of 1,200 solid white squares of 6different sizes, and random, in plane, angular orientations (FIG. 3,top), and used it to train the network. First, randomizing all gridpixel values initialized the network's output grid images. Forillustration, the surface of the cylinder is shown as a large squarerectangle (starting conditions in FIG. 3), cut along the height, fromtop to bottom, and laid flat (circumference, horizontal; height,vertical). Remember that the left and right borders of this rectangularrepresentation are attached to one another in the neural network. Thefirst training image (“a”) was randomly selected, randomly rotated(importance of which explained below), and presented to the network,with the output grid being modified (trained) by this image. Then,another training image (“b”) was randomly rotated and presented; theoutput grid was modified again, etc. As will be seen below, theparticular order is irrelevant because the alignment results becomeorder independent when the learning rate drops below threshold.

Training was continued for 30,000 cycles. For every cycle of imagepresentation, all the grid images were made to morph slightly towardthat of the presented image. For that reason, even though initialized ina random pixel state, the observed final grid images (output A) finallyresembled the input squares. However, the amount of morphing of outputgrid images was not the same everywhere. It decreases as one moves awayfrom the “winning image,” defined as the grid image that most closelyresembles the presented image {grid image with lowest Euclidean distanceto presented image, with the images represented as vectors). Thisdecrease is made to follow a Gaussian distribution. We decreased thestandard deviation of this distribution in successive training cycles.The level of morphing was also made to decrease (linearly) as thenetwork training progressed, in order to fine tune the results (set byLO=0.1; this higher value for the learning rate works well withsynthetic data sets because there is less heterogeneity). Theseconditions allow the grid to become ordered: first globally, thenlocally, as is evident when observing the following in FIG. 3:

1) Moving horizontally across output A, from any starting position, thesizes of the white square images remain constant, but the angularorientations rotate uniformly.

2) Moving vertically across output A, from any starting position, theconverse is true: the image sizes increase or decrease uniformly, butthe angular orientations remain constant.

Therefore, the network separated angular variation and size variationalong orthogonal directions of output A. From the above observations weconcluded that, because of the local and global ordering of output Aimages—regarded as bins, with sorting of the training set images intobins of greatest resemblance—the data set gets classified according toboth size and angular orientation. The vertical position indicatesrelative size, and the horizontal position, relative angularorientation. We performed the mapping by using, as a similaritycriterion, the Euclidean distance between the vectors that the imagesrepresent; the shorter the distance between vectors, the greater thecorresponding similarity.

Output B shows the average contents of each bin, after the mapping. Eachof the 6 size classes is observed to fall into a distinct horizontal rowof images. The largest and smallest sizes, representing the endpoints ofthe data set's principal bounded variation (size), were mapped to theupper and bottom most portions of the cylindrical grid (because thecylinder's top and bottom are its topological boundaries). The relativeangular orientations of the images mapped to each column are indicatedby the azimuthal angle axis at the bottom of output B. Because the testimage shapes (mapped to the output grid “bins”) rotate uniformly aroundthe circumference of the cylinder, the axis is calibrated in 90°/15=6°increments. That is, images mapped to the first column of the cylinder(the left most column, defined as column 1) have relative angles of 0°.Images mapped to the second column have relative angles of 6°, and soon, down to the last column where the mapped images have relative anglesof 84°.

We compared the known azimuths of the test images to the angles obtainedfrom output B, determining that angle assignments were correct to within6°. We obtained proportionately greater accuracy in rotationalalignment, by increasing the circumference of the cylinder. For example,on a run with a circumference of 90 neurons, the azimuths for the testimages calculated using SORFA did not deviate more than 1° from theirknown values. The only limiting physical factor in resolving theazimuths appears to be the number of pixels in the test images,themselves.

To summarize, we started with a randomly oriented, heterogeneous mixtureof square images. Using SORFA, with a cylindrical topology, we correctlydetermined the azimuths of all 1,200 images. In addition, SORFAcorrectly classified the 6 square sizes in the training set into 6distinct rows.

3.1.2 Evaluation of Azimuthal Angle Classification

The following “test run” determined the consistency of the azimuths.First, we rotationally aligned the test set, by applying the relativeazimuth for each test image, and then used this aligned test set totrain the network, just as above. After training, the mapped images fellinto two adjacent columns (test run in FIG. 3). This was an optimalresult.

This follows because, for example, if images from 0° to 6° are mapped tocolumn one and images from 6° to 12° are mapped to column two, then twoclosely aligned images at 5.9° and 6.1°, respectively, would be mappedto separate columns, even though the separation is only 0.2°. Theprobability of this, or the equivalent, occurring in a field ofthousands of images, is almost a certainty. Therefore, mapping of alltest run images to a single column is highly unlikely.

Consistent results also were obtained with other test image symmetries,when applying azimuthal angle classification (for example, rectangular,triangular, pentagonal, hexagonal, heptagonal, and octagonal test sets;data not shown). A general rule emerged from these experiments: the testimages rotate around the cylinder by (360°/image symmetry). For example,hexagons, with 6 fold symmetry, rotate by (360°/6)=60° around thecircumferential axis of the grid. This is expected, because a rotationof 60° brings a hexagon back to its original orientation, with thevariation being unbounded, representing a full turn around the cylinder.Therefore, for test images of n fold symmetry we labeled the azimuthalaxis of the cylinder in [(360°/n)/(circumference)] degree incrementsfrom column to column. Conversely, the symmetry of the data set may beempirically determined by dividing the number of degrees of rotation ofthe grid images around one full turn of the cylinder into 360°.

3.1.3 Influences of Noise on Azimuthal Angle Classification.

We added various noise levels to the synthetic squares described aboveand performed azimuthal angle classification. When we used the samestarting parameters described above, no effect on the precision oraccuracy of the azimuthal angle determination was observed for noiselevels yielding SNRs as low as 0.07; with higher noise levels, we couldstill obtain good alignment by changing the starting parameters.

We next applied SORFA to a more realistic test set of randomly rotatedprojections of the atomic model of AQP1 (Sui et al., 2001), with noiseadded (top of FIG. 4). The model was split down the middle to generateprojections that represented the cytoplasmic and extracellular domainsas viewed head on. We added varying levels of noise, and generated 1,200randomly rotated representatives for each of the two sides. These imageswere then used as a test set to train SORFA, with a cylindricaltopology, for 5,000 cycles at an initial learning rate of 0.1.

As is evident in output B of FIG. 4, the two projected sides wereperfectly separated around the cylinder's top and bottom. After bringingthe images into alignment using the output B azimuths, the respectiveaverages (aligned training set averages in FIG. 4) wereindistinguishable from the original images, before adding noise. Themapped images in the test run were nearly completely confined to twocolumns (test run in FIG. 4). Note that the numbers in the lower-righthand corners indicate the number of images mapped to, and averaged, ineach bin. When adding noise down to an SNR of 0.25, the bandwidth of thetest run did not increase. Because the untilted AQPO images have an SNRof ˜1, we ignored the effects of noise in the analysis of this data set.

3.1.4 Structural Difference Classification Using the Cylinder

In the above experiment, the cylinder perfectly separated the twodomains of the AQP 1 x-ray model, with noise added down to a SNR of0.25. After obtaining perfect separations using a wide variety of noisytest images, ranging from human faces to electron micrographs, weconcluded that the cylinder is inherently robust at classifying a noisytraining set consisting of only two noisy classes. As the number ofnoisy classes increases beyond two, however, the cylinder's ability toproduce homogeneous classes rapidly deteriorates. Therefore, when thereare more than two noisy classes comprising the training set, analternate method should be used. We experimented with the N-CUBE as apossible tool for such classifications.

3.1.5 Structural Difference Classification Using the N-CUBE

We performed N-CUBE SOM tests for values of N=0, 1, 2, 3, and 4 toevaluate the network's ability to handle up to 8 test classes. Eachclass consisted of a variable number of elements, each a differentmodification of a single image: the “‘generator image.” Themodifications only altered orientation and clarity, not structure. Wemodified the images by, both introducing a limited random rotation(5°≤θ≤5°, and adding Gaussian white noise. Two of the 8 generator imagesare the cytoplasmic and extracellular projections of the x-ray AQPImodel. The other 6 were formed by imposing various symmetries (3 fold to6 fold) on the first two.

The purpose of introducing noise and limited random rotation is to tryto make structural classification more difficult through variations thatdo not alter class membership, only spatial orientation and clarity.FIG. 5 shows the generator images of the eight classes in the leftcolumn, and one example from each of the corresponding noisy, rotated,test classes in the right column. The (5°≤θ≤5°) random rotationalvariation made the image alignments more comparable to those of a noisyEM data set. Our choice of applied random variations was based on ourobservation that the x-ray AQPI test set, was easier to align than theactual AQP0 micrographs. For example, the 4S neuron circumferencecylinder aligned 99% of the AQPI test images to within ±I 0 (see above),whereas 82.7% of the AQPO images were to within only ±2.0° (see below).

We also trained the N-CUBE SOM for N=0, 1, 2, 3, 4, using combinationsof 1 to 8 test classes, with class membership size varying from 1 to100, with varying amounts of noise. We first carried out tests withoutnoise or rotation. In nearly every instance, the following two casesdescribe the behavior of the test classes near the vertices. Thisbehavior helps the operator to locate homogeneous subsets of theprincipal (largest) test classes by looking at the images mapped tonear, or at, the vertices.

CASE 1: The number of test classes is≤than the number of vertices: eachand every member of the same test classes maps to the same vertices (ora position within their immediate neighborhood). For example, thetop-right panel of FIG. 5 shows output B when two test classes, A and B,are analyzed in a 2-CUBE SOM (four vertices). Class A (49 members) andclass B (53 members) map to diametrically opposite vertices. When thedata set contains only two classes, they nearly always map todiametrically opposite vertices. The middle-right panel of FIG. 5 showswhat happens when two additional test classes are present, making atotal of four. Class C (42 members) and class D (59 members) map tovertices, as well as class A (49 members) and class B (S3 members).

CASE 2: The number of test classes is>than the number of vertices: onlythe test classes with largest memberships map to the vertices (theprincipal classes). The bottom-right of FIG. 5 shows the case when adata set, which contains 8 test classes, is analyzed in a 2-CUBE SOM (4vertices). Only the 4 largest test classes map to the 4 vertices: ClassD (59 members), class B (53 members), class E (51 members), and class G(51 members). The 4 smaller test classes map to unpredictable positionsalong the edges. Therefore, in order to position a maximum of 2^(N)principal test classes around the vertices, an N-CUBE SOM should beused. Each shape can.be thought of as an endpoint of a variation whoseboundaries are the shapes themselves. The principal shapes may map tothe vertices because they seek the largest mapping domain.

As noise is gradually added, and the limited random rotation isintroduced, a fraction of the mapped images begin to drift away from thevertices. This probably is because the significant variation introducedby noise breaks down the well defined boundaries between classes.Eventually, with enough noise added, the vertex classes becomecontaminated with members from other test classes. The effect ofincreasing N on vertex class homogeneity is shown in FIG. 1 G, revealingthat vertex class homogeneity progressively improves as N increases from0 to 4, and possibly beyond. For example, when N=4 and the SNR=0.32, thevertex classes are still perfectly homogeneous.

3.2 Analysis of AQPO Images

With the understanding that relative azimuthal angles may be read off ofthe circumferential axis of the cylinder, and that principal classes maybe read off of vertices of the N-CUBE, we progressed to classifyinguntilted AQP0 images according to their azimuthal angles and principalstructural differences.

3.2.1 Azimuthal Angle Classification of Symmetrized AQP0

3,361 untilted images of freeze fractured AQP0 particles were used. Thisdata set consists of views of the cytoplasmic and/or extracellularfaces, and in varying conformations. Symmetry was imposed to attenuateany image heterogeneities caused by metal layers of variable thicknessdeposited on the fracture faces, and to improve alignment [also seeBoonstra et al., (1994)]. This use of C4 symmetry was based on theanalysis of Eigen images described by Zampighi et al. (2003).

A cylinder of 45 grid image circumference, was used to calibrate theazimuthal axis in 90°/45=2° increments. The network was trained for300,000 cycles with a learning rate of 0.001. Output B is displayed inFIG. 6. The final grid images in output A (not shown) were nearlyindistinguishable from the average of the aligned images. As with thetest images, we observed that the angular orientations of the gridimages uniformly cycled around the cylindrical grid. The images mappedwith the smallest images at the top, and progressed to the largestimages at the bottom.

Relative azimuths of all 3,361 images were read from output B, by notingto which cylinder column bins the images were mapped. Next, a test runon the precision of the azimuths, gave the results shown at the bottomof FIG. 6. Here, 82.74% of the images fall into two columns. Therefore,82.74% of the images were precise to within 2 4° for the two SORFA runs(bottom of FIG. 6).

We repeated the alignment procedure with a higher learning rate of0.0025, to ascertain its effect on the quality of the test run. Withthis higher learning rate, 65.40% of the images were precise to within 24°. Thus, the lower learning rate produced better alignment. Thisprobably ensues because of the large amount of heterogeneity of the dataset. A lower learning rate weakens the influence of training imagesduring training, thereby reducing any bias from the early part oftraining. A learning rate lower than ˜0.001, however, had no effect onthe standard deviation of the test runs, even with increased numbers ofiterations out to 3 million (up to a ten-fold increase). Reducedlearning rates also had no effect on the precision of the azimuthalangles obtained between independent SORFA runs. Therefore, it was notnecessary to go below a learning rate of 0.001 because the threshold hadbeen reached where a reduced influence of the training images had noeffect in improving alignment. A second run at the 0.0025 learning rateallowed a comparison of the azimuths between the runs, to ascertainangular reproducibility between the two 0.0025 learning rate runs. Wearbitrarily selected particle #124 as a 0° reference. The averageangular deviation of corresponding images between the two runs was only±2.47°, close to the value predicted from the test run. The alignmentresults were, therefore, essentially independent of the order of theimages used during training.

3.2.2 Structural Difference Classification and 3D Reconstruction ofSymmetrized AQP0

After azimuthal variation had been largely eliminated from the AQP0 dataset by applying the cylindrical topology, the vertex classes of the 2DAQP0 data were partitioned using an N-CUBE topology. We used the alignedAQP0 images to train the N-CUBE SOM separately for values of N=1, 2, 3,in order to cluster a maximum of 2^(N) vertex classes, respectively. Thesame number of training cycles (300,000) and initial learning rate (0.1)were used in the following runs.

For N=1, we used a 225×1 output grid. After training, we generated twovertex classes by taking the images from the two vertices and theirimmediate neighbors. Enough neighbors were used to bring the numbers inthe two partitions to 168 for the left partition, and 171 for the rightpartition.

For N=2, we used a 15×15 output grid in order to keep the number ofneurons the same as in the N=1 case. After training, we generated 4classes by taking the images from 3×3 arrays at the vertices. Thesearrays were used because they are the smallest that contained anadequate number of images to generate 3D reconstructions (LL, 174; LR,167; UR, 182; UL, 179 images, respectively).

For N=3, we used a 6×6×6 output hypergrid. This grid was selectedbecause, of all the N³ networks, it is closest in number of neurons to225, the hypervolume of the N=1 and 2 cases above. After training, wegenerated 3D reconstructions from all 8 vertex classes. We selected2×2×2 arrays from the vertices in order to harvest a number of imagesadequate for 3D reconstructions (136, 148, 158, 159, 164, 168, 174, 181images, respectively).

When we compared the class averages shown in FIG. 7 with the onesobtained in a previous study (Zampighi et al., 2003), the output Bvertex classes produced by SORFAC exhibited very similar overalldimensions and shapes to the classes calculated by the alignment throughclassification procedure (Zampighi et al., 2003). The particles rangedfrom 92.1-112.0 Å for N=1, 93.9-106.3 Å for N=2, and 96.0-106.0 Å forN=3. There is a correlation between an increase in Nanda decrease in theangstrom range.

For N=2, three of the vertex classes of output B exhibited similaroctagonal shapes but different sizes (Class A, C, and Din FIG. 7). Thefourth vertex class exhibited a smooth tetragonal shape (Class B). Asecond independent run also produced a smooth tetragonal vertex class.This time, however, two vertex classes (Class E in FIG. 7) of very muchdifferent looking octagons appeared, with nearly 90° and 180° internalangles. These octagons had been located at a non vertex position in thefirst run. On the contrary, the octagonal vertex class found in thefirst run was located at a non vertex position in the second run. TheN=3 hypercube, however, captured all of the above shapes. It was moreencompassing, capturing size and shape categories common to all of ourindependent runs using N=1, 2, and the cylinder (FIG. 7). Apparently,more than 4 classes are competing for the 4 vertices in the N=2 case,and more than 2 in the N=1 case.

After obtaining the vertex classes, the corresponding tilted views wereused to reconstruct the 3D molds of AQPO, as in the previous study(Zampighi et al., 2003). These (shown in FIGS. 8-10) appeared as cupswith external convex and internal concave surfaces. Imprints, whichrepresent the molecular envelopes of the particles, were calculated fromthe concave surfaces of the molds.

3.2.1 Populations A and B of 3D Reconstructed Molds

The molds' dimensions were similar, but their structural featuresdiffered. When we examined reconstructions corresponding to classesisolated with N cubical arrays for values of N from 1 to 3, we observedthat two populations of molds could be identified. Population A ischaracterized by four large elongated depressions in the concave side ofthe mold, depressions that are separated by 24.2-27.5 Å. Adjacentdepressions are at right angles to each other. Population B ischaracterized by four ring-like structures at the base of the concaveside of the mold separated by 47.5-51.7 Å, and by four elongateddepressions (“footprints”) that are separated by 30.0-38.0 Å. An exampleof one or more members from each population is shown in FIGS. 8 and 9,respectively.

3.2.4 Comparison of Populations A & B to 2D Classes

We found that the 2D AQP0 classes C and D (FIG. 7) yield molds that aremembers of population A. Population B molds are produced by classes A,B, and E. The smaller 2D image classes correspond to population A molds,and the larger 2D classes to population B molds. Classes of group Fyielded molds that belonged to either population A or B.

3.3 Comparison of Population A to the Extracellular Domain of the AQPIX-Ray Model

In the two views of the mold shown in the left panel of FIG. 8, fourlarge oval like depressions (seen as depressions in 3D viewing) arepresent in the concave side. Any two immediately adjacent oval likedepressions are at right angles to each other and are separated by 24.2Å. These features correlate well with the four oval like elevations seenon the extracellular side of the AQPI x-ray model (FIG. 8). Theseelevations in the x-ray model are believed to correspond to projectingpolypeptide loops that connect to alpha helices (see overlay at bottomleft in FIG. 14). Adjacent loop peaks are separated by 24.4 Å.

Like the depressions in the mold, each oval loop density is orthogonalto both of its immediate neighbors. The four depressions in the moldcorrespond to four elevations in the imprint (right panel of FIG. 8).Two of the four peaks in both the imprint and x-ray model are marked byarrows in FIG. 8. The distances, shapes and orientations of theprincipal depressions in the mold correlate well with the distances,shapes and orientations of the principal elevations of the extracellularside of the AQPI x-ray model. By taking cross sections of the molds, wedetermined that the internal length and width dimensions of the moldaverage 62.4 Å. The x-ray model dimension of 64 Å is also similar to thevalues we calculated for the molds.

3.4 Comparison of Population B to the Cytoplasmic Domain of the AQPIX-Ray Model

Four examples of molds corresponding to population Bare shown in FIG. 9.These reconstructions correspond to a vertex class that mapped oppositeto the example used for population A. The observation that classesrepresenting opposite sides of AQP0 map to diametrically oppositevertices of the planar output grid was known (Zampighi et al, 2004). Thecytoplasmic side of the AQP1 x-ray model is shown at bottom left), forcomparison. Four outwardly pointing elevations (“‘fingers”) can be seen.The endpoints of the thin black line indicate two of the elevations.These correspond to the amino termini of the protein monomers, with 51.0Å being the distance between them. These elevations correlate well withthe rings that are found in the mold at left center (indicated byendpoints of thin black line) that are separated by 50.0 Å. One canimagine the “fingers” fitting into the rings. The inside diameters ofthe rings are close to the diameters of alpha helices (FIG. 11). Thecloseness of the orientation match of the amino termini, and thelocation of the rings and holes with respect to the mold, is illustratedin FIG. 10. Here we fit the cytoplasmic half of the AQP1 x-ray model(docked into the imprint of Zampighi et al., 2004) into our mold. Theamino termini of the x-ray model project directly through the rings ofthe mold and out through four holes in its back. In the case ofpopulation Bit is easier to see the details in the molds than in theimprints. For that reason, the corresponding imprints are not shown.

The lengths and widths of footprints are consistent with the dimensionsof the carboxylic tails in the x-ray model. Shown in FIG. 11 are thefootprints for N=2 and 3; they are 20.8 Å and 20.5 Å, respectively, inlength, and 8.5 Å and 7.4 Å, respectively, in width. The widths of theholes and of the depressions in the molds yielded by the classesisolated with N cubical arrays for values of N=1-3 are also very closeto the diameters of alpha helices. The footprints are found at allintermediate positions directly between the rings, and exhibitdifferences in orientation.

When molds are reconstructed from classes isolated by using N cubicalarrays with higher values of N, both the footprints and the ring-likestructures in the molds are significantly more prevalent. Perhaps thisowes to the greater homogeneity of the vertex classes at larger valuesof N. When N=1, neither of the molds has footprints/rings. When N=2 anaverage (over several runs) of three in eight molds havefootprints/rings. When N=3 an average (over several runs) of six ineight molds have footprints/rings.

As can also be seen in FIG. 10, another characteristic of population Bmolds is that the central portion of the floor of the concave side ofthe mold presents a dome shaped region of positive density, which wouldcorrespond to a dome shaped region of zero density in the imprint. Inthe AQPI x-ray model there is a “crown” made up by the amino andcarboxylic tails with a zero density inside the crown. By taking crosssections of the molds, we determined that their internal length andwidth dimensions average 64.9 Å. The x-ray model is similar at anaverage of 64 Å.

3.5 The Complete Channel

Freeze fracture exposes structures representing only half of thechannel. Therefore, we needed to sum the heights of the respective halfchannels to obtain the complete length. This calculation yielded a meanvalue of 67.2 Å.

We report a method of analysis of the unsymmetrized AQP0 data set inExample 4.

4. Discussion

4.1 Rationale of Image Reconstruction Procedures

Our first goal was to develop a reference free technique for determiningrelative rotational orientations of 2D EMs by exploiting the SOM'sremarkable topology, metric conservation, and parallel processingarchitecture. It is known that, when the SOM maps high dimensional inputdata (e.g., a set of EMs) onto an output grid, it tends to conserve thedata's topological structure and metric. In particular, given the limitsof the topology of the network, it tends to conserve the topology of theprincipal variations in the training set. We increased the weight ofangles, as principal linear variables in the data set, by extensivetraining of the SOM with images in random orientations. This tends toestablish a linearly calibrated azimuthal axis in the mapped to outputspace and to eliminate angular bias during training. This follows fromKohonen's demonstration that inputs that occur with greater regularityduring training, map into larger output domains (Kohonen, 2001). Becauseangular variation is unbounded, an unbounded coordinate axis was needed.Otherwise, the numbering system along the axis would be meaningless,because any azimuth augmented by the angle of symmetry must restore thesame axial position.

In view of the above, we attempted to match the topology of the grid tothat of the data. First, consider the principal topologicalcharacteristics of the data when using the RCT method. Because theparticles are largely free to rotate in the plane of the bilayer, therewill be topologically unbounded rotational variation in the data (a fullrotation is unbounded). Also, because the particles are either “flipped”one way or the other in the bilayer, this type of variation istopologically bounded (by the two domains of imaged particles). Othertypes of variation include differences in molecular conformation, whichalso appear to map to the network's vertices during structuraldifference classification.

We anticipated, that if the above matching was achieved, the network, inattempting to conserve the topology of the data on the grid's surface,would map unbounded variations around topologically unboundeddirections, and bounded variations along the grid's topologicallybounded directions. By matching the topology of the output grid to thedata in this way, we expected the network to establish an intrinsiccoordinate system on the grid's surface.

One of the coordinates would be our first unknown, the azimuth; theother would be our second unknown, the orientation in the plasmamembrane. The first piece of information would be precisely what isneeded to generate 3D reconstructions. Kohonen (1982) demonstrated thatthe SOM maps similar data set images to neighboring regions on theoutput grid. For that reason, we expected azimuthal changes to bemonotonic around the circumference of the cylindrical grid. Whenreferring to the network's system of cylindrical coordinates, we callthe unbounded axis (the axis wrapped around the cylinder circumference)the “azimuthal axis” (bottom of output B in FIG. 3), and the boundedaxis the “height axis” (left edge of output B in FIG. 3).

With the above expectations, we implemented an output grid topology thatis both bounded and unbounded. One such topology is the above mentionedcylinder, unbounded around any circumference, and bounded along anydirection perpendicular to it (the cylinder's top and bottom are itstopological boundaries). As we expected, after the network was trainedusing a cylindrical grid, images corresponding to a particular relativeazimuth were mapped appropriately around the cylinder's circumference,and those corresponding to the data set's principal bounded variation,size, were mapped along the cylinder's height. In other words, azimuthand size were mapped in a cylindrical coordinate fashion.

This technique, making use of a cylindrical output coordinate system forreference free alignment of 2D particles, is referred to as,“Cylindrical Topology-Self Organizing, Reference Free Alignment” (CTSORFA). The cylinder, however, is not the only topology that meets theserequirements, merely the simplest. A unique characteristic of SORFA isits exclusive dependence on the SOM for reference free rotationalalignment of translationally aligned, 2D particle images. The learningcoefficient, number of iterations, learning radius, and height andcircumference of the cylinder, were all varied to probe for improvementsin the final mappings, with results summarized in section 2.5.

The SORFA method contrasted with the alignment through classificationmethod in both speed and lack of operator dependent references. Itentailed only hours of computer processing as opposed to months. Thisincluded the extra time that was necessary to arrive at the best initiallearning rate, and optimal circumference/height ratio of the cylinder.FIG. 15 shows 3 test runs (3 left most) under the same conditions (45neuron×45 neuron cylinders at 300,000 iterations, and a learning rate of0.001) using the same 3,361 AQPO data set, but with three differentmodes of alignment (from left to right): 1: using initial non-alignedimages; 2: using SORFA aligned data {alignment took ˜2 hours); 3: usingMSA/MRA aligned data (alignment took ˜2 weeks using 6 rounds ofMSA/MRA). FIG. 15 also shows the test run of an independently producedMSA/MRA aligned set, also using the same test run conditions (farright). The data set was nearly identical, but was filtered to 8 Åinstead of 24 Å for the alignment and classification stages. The imagesalso used very tight masks, and consisted of 3,366 images. Operatorerror likely resulted in the double peak in this MSA/MRA aligned testrun. The excessively tight masking resulted in the images being mappedto the upper half of the cylinder, as can be seen in FIG. 15 (farright).

We have demonstrated that SORFA accommodates both the artificiallygenerated test sets (of various symmetries) and symmetrized andunsymmetrized EMs. SNRs for alignment as low as 0.25 do not requireinitial parameter alterations or materially affect the accuracy andprecision of the azimuths. After largely removing rotational variationwith CT SORFA, the topology is switched from cylindrical to NDhypercubical, in order to partition more homogeneous clusters(representing the different conformations and views of the object beingstudied), and to control the number of the vertex classes generated.Values of N=1, 2, and 3 were used. Further, it is easy to evaluateparticle alignment, and no assumption need be made about a referenceimage.

Because of SOM's metric preserving characteristics, highly dissimilarclasses of input image are clustered to maximally separated regions ofthe hypergrids. Because these regions in the N-CUBE are located atdiametrically opposite vertices, the N-CUBES clustered these classeswithin the vicinity of the 2^(N) vertices. The strength of SORFA, inmapping the principal classes to the vertices, was demonstrated evenwhen the test images were not perfectly aligned.

4.2 Interpretation of Results

4.2.1 Relating to Procedures and Earlier Findings

The results of 3D reconstructions from partitions, taken for N=1, 2, 3,are shown in FIGS. 8 and 9. A reason why partition homogeneitysignificantly increases for higher dimensional hypergrids (FIG. 1G) maybe the decreased data compression when mapping high dimensional dataonto yet higher dimensional hypergrids. When multiple differences exist,such as macromolecular conformational variations, perhaps they canbecome better separated in higher, rather than lower dimensional spaces,by virtue of additional degrees of freedom. Perhaps this accounts forthe lesser contamination in higher dimensional vertex partitions. Adisadvantage is that, in practice, one cannot determine whether allprincipal classes are represented at the vertices, or whether N isneedlessly high.

The new features in the molds are located precisely where previousvolumes were unaccounted for (Zampighi et al., 2003). These volumes arejust outside the envelope of the cytoplasmic domain, as detennined byZampighi et al. (2004), where docked amino and carboxylic termini of thex-ray model project (FIG. 10). We expect to see holes and footprints inthe molds, corresponding to opposite densities of amino and carboxylictermini and, indeed, they are seen. The footprints, however, areindeterminately located between the holes/rings at a mean distance of25.2 Å from the central axis of the mold. For example, in the mold ofthe unsymmetrized alignment (see FIG. 14 and Example 4), we see thefootprint in an intermediate position, and at a different angle. Iffootprints are depressions made by carboxylic tails, one way to accountfor the orientation change would be the ability of carboxylic tails torotate and/or change conformations (indicated in FIGS. 11 and 14). InFIG. 11, the carboxylic terminus would be proximal to the amino terminusof the adjacent monomer.

4.3 Future Applications

Future applications of SORFA to SOMs other than the Kohonen SOM, used inthe above illustrations, could generate better topographic orderings;for example, fuzzy Kohonen clustering networks (FKCN) (Pascual et al.,2000) and KerDenSOMs (Pascual Montano et al., 2001), and ParameterizedSelf-organizing Maps (PSOM), where continuous mappings may be attained(Walter an Ritter, 1996). Another application would use hypersimplexarrays, where the number of vertex classes would not be limited tomultiples of two, but could be set to any positive integer. Finally, theEuler angles needed for angular reconstruction of cryo-electronmicroscopical data may be obtainable from other topologies like, forexample, spherical, hyperspherical, toroidal, hypertoroidal, etc.,depending on the topology of the cryo-data-set and the symmetrie(s) ofthe 3D object(s) being studied. SORFA could also be parallel processed,not simply emulated using serial computing. Then, all the neurons couldbe compared and updated simultaneously, not just in sequence. Such aparallel processed architecture could be implemented on a microchip. Itpromises to speed processing by several orders of magnitude. Forexample, processing on a 45×45 neuron cylinder, with 2,025 neurons,would expect to run at least 2,025 times faster, when parallel processedthereby decreasing processing times to a few seconds, or less.

Beyond their independent applications, these new techniques also promiseto extend the utility of the familiar x-ray and electroncrystallographic space filling, ribbon, and cylinder and loop,reconstructions, as well as those of other advanced imagingmethodologies. A main advantage is that crystallization is not employed.Consequently, many objects can be examined in close to their nativeconformations. Among many other anticipated applications, the newtechniques could significantly impact neurophysiology and drug designand discovery. Further, being independent of object size, they mightfind applications from very large macromolecular assemblies, such asribosomes, to medical patient diagnostics where repetitive movements andonly tens of milliseconds exposures are involved, for example, highspeed cardiac Mill.

The present invention is more particularly described in the followingexamples which are intended as illustrative only since numerousmodifications and variations therein will be apparent to those skilledin the art. The fol lowing examples are intended to illustrate but notlimit the invention.

EXAMPLE 1 Locating the Vertices of a Tesseract and Calculating DistancesAlong a Cylindrical Surface

Locating the hypercube vertices for the cases N=1, 2, and 3 isstraightforward, because lines, squares, and cubes are easilyvisualized. However, visualization of the N=4 tesseract geometry is moredifficult. In spite of this, the vertices are easily located by simply“counting neurons.” The neurons (identified by their coordinates in 4space) are first ordered using a simple raster convention. For example,a 3×3×3×3 hypercube with 34=81 neurons are numbered as follows in4-space: 1: (0,0,0,0), 2: (1,0,0,0), 3: (2,0,0,0), 4: (0,1,0,0), 5:(1,1,0,0), 6: (2,1,0,0), 7: (0,2,0,0), 8: (1,2,0,0), 9: (2,2,0,0), 10:(0,0,1,0), . . . , 81: (2,2,2,2). For a tesseract of length, L neurons,the 16 vertices are located by counting off the following neurons (seeFIG. 2 for a 2D projection of a tesseract).

V1 = 1 V2 = L V3 = L (L − 1) + 1 V4 = L2 V5 = (L − 1) L2 + 1 V6 = (L− 1) L2 + L V7 = (L2 − 1) L + 1 V8 = L3 V9 = 1 + L4 − L3 V10 = L4 − L3 +L V11 = L4 − L3 + L2 − L + 1 V12 = (L2 − L + 1)L2 V13 = (L2 + 1) L2 + 1V14 = L4 − L2 + L V15 = (L3 − 1) L + 1 V16 = L4

For the 3×3×3×3 hypercube L=3. Therefore, the 16 vertices are locatedat: VI=1, V2=2, V3=7, . . . , V16=81. After training the 4 cube,˜(150-200) images are harvested from the images mapped to each of the 16vertices and their immediate proximity (to bring the number up to the150-200 range) of the output tesseract, and used to generate 16 3Dreconstructions. Faster camera sampling rates (>1/3.1 Å), andimprovement in other factors required for better resolution attainmentincluding, but not limited to, decreased spherical aberration(<2×106nm), decreased electron wavelength (<0.00335087 nm), reduced defocusmaxima [<(200-300) nm], etc., may require larger numbers of images to beharvested in order to achieve the best results.

When calculating distances from the winning image on a cylindricalsurface, the network only considers shortest distances. We represent apoint on the surface of a cylinder of a given height, H, andcircumference, C by (a, h) where “a” is the azimuthal coordinate and “h”is the height coordinate (see Results). The formula for the shortestdistance on the surface of a cylinder between points (a1, h1) and (a2,h2) is:

min{[(a1−a2)2+(h1−h2)2]1/2, [(C−|a1−a2|)2+(h1−h2)2]1/2}.

EXAMPLE 2 The Importance of Choosing a Low Value for the InitialLearning Rate

Lower values of the initial learning rate allow the output grid imagesto gradually adapt to the data, so that each image presentation has onlya small influence on the output grid. The advantage is that the grid isnot forced into an initial bias during early training. A sufficientlylow value of the learning rate eliminates training bias because thealignment results become independent of the order of the images used totrain the network. This is because a continued reduction of theinfluence of the training images during training (a continued reductionof the learning rates, with a corresponding increase in numbers ofiterations) has no effect on the alignment precision between independentruns. The choice of the right learning rate is critical for globalalignment of the training set, because values of the learning rate thatare too high cause small amounts of grid image rotation along the heightof the trained cylinder (particularly with much heterogeneity in thetraining set). Contrariwise, one wants all rotational variation to beconfined along the azimuthal axis.

The network will be over trained or under trained, if the initiallearning rate is too large or too small, respectively (withoutcompensating by decreasing or increasing the number of iterations,respectively). When it is optimally trained, the test run's histogramhas the narrowest distribution. We have empirically observed that, whenthe network is optimally trained, the grid images in output A vary fromvery similar, to nearly indistinguishable from the aligned average ofall images.

When the network is over trained, all the grid images no longer resemblethe aligned average of all images combined; instead, some are distinctlooking, aligned subclasses. During training, the grid images move froma random state to one resembling the average of all aligned images, whentraining is complete. Moreover, with over training, they differentiateinto distinct subclasses, plainly evident upon visual inspection.Inspecting the grid images in this way is useful to quickly assessquality of alignment before completing the test run (see Materials &Methods). The test run, however, is the definitive tool for evaluatingqualify of alignment.

EXAMPLE 3 Angular Deviation from Linearity as a Function of the TotalNumber of Training Cycles

For images with 4 fold' symmetry, for example, the range of azimuthalangles is 0° 90°. Consider a series of 90 boxes, where all of the boxesare labeled in the following way: Box 1, 0°-1°; Box 2, 1°-2°; etc., downto the last box; Box 90, 89°-90°. Think of a ball being dropped into abox of a given angle range, whenever an image is presented to thenetwork, during training, with that given azimuthal angle range. Aperfectly linear distribution of presented angles during training wouldthen arise whenever exactly the same number of balls was in each of the90 boxes, at the end of training. This number would be equal to theaverage. That is, the total number of images presented during trainingdivided by 90. In practice, however, because the distribution is random,some boxes will have more balls than others, which accounts for thenon-linearity of the angle distribution. Therefore, the deviation fromlinearity is the deviation from the mean. An overall measure of thisdeviation from linearity is the standard deviation of the distributionof all deviations from linearity across all boxes.

The Kohonen network, however, is not concerned with the absolute valueof the angular deviation from linearity, but its percentage of thewhole. This is because, when an image makes up a larger percentage ofthe images used for training, the image's influence in the final mapwill be greater. Correspondingly, if the deviation from linearity is asmaller percentage of the totality of presented angles, the influencewill be lesser. In view of the above, we define the measure of influenceof nonlinear angle presentations to be 100% times the standard deviationof the departures from linearity across boxes, divided by the totalnumber of balls.

In FIG. 12, the percentage deviation from linearity is plotted againstthe total number of training cycles. Note that between ˜100,000 and˜300,000 training cycles, the hyperbolic looking curve essentiallybottoms out. At 300,000 training cycles, the deviation from linearity is0.18%. Since beyond 300,000 training cycles there is not much reductionof non-linearity we chose 300,000 training cycles in our final alignmentruns.

EXAMPLE 4 Classification and 3D Reconstruction of Unsymmetrized AQPO

The unsymmetrized AQPO data set was aligned in a 45×45 neuron cylinderwith a learning rate of 0.001. Images in output A rotated 180° acrossthe circumference of the grid, indicating a majority has 2 fold symmetry(not shown), with output B results shown in FIG. 13. The 2 fold symmetrymay be partly due to the elongation of many 2D untilted images, as aresult of their tilt angle not being exactly 0°. The aligned images werethen used to train SORFA with an N=2 hypercubical topology, with outputB results shown in FIG. 14. To quantify and refine the precision ofcalculated azimuths of the unsymmetrized AQPO data set, the clustersclosest to the 4 vertices were selected. For example, the upper rightvertex class was selected and used to retrain SORFA, but with a cylindercircumference of 100 neurons, a cylinder height of 10 neurons (best testrun), a learning rate of 0.001 and 300,000 iterations. The imagesrotated 90° around the grid, indicating that this subset had an overall4 fold symmetry. The grid images in output A resemble the alignedaverage (FIG. 13). The 3D reconstruction corresponding to this alignedclass is shown in FIG. 14 (top right). This mold has characteristics ofpopulation B, except for the presence of rings. The four footprints areseparated by 36.8 Å. The inside length of the mold is indicated by theblue line and is 64.2 Å.

EXAMPLE 5 Hyperalignment and 3D Reconstruction of Symmetrized AQPO

We also test d a method to “refine” the alignment of the images.Hyperalignment is the re alignment of the aligned images, or startingimages (in order to prevent accumulation of aliasing and/or roundingerrors due to successive rotations of the same image), in order torefine the azimuths, but using subsets generated from SORFA (such as avertex partition) that have greater homogeneity. The subsets areselected from within neighborho9ds anywhere in the N cube SOM'shypervolume, after structural difference classification. As an exampleof this procedure, we took the vertex partitions from the N=2 case,above, ˜(167 182) images and individually realigned them with SORFA (seeFIG. 16). Azimuthal axis resolution was increased by increasing thecylinder circumferences to 100 neurons (0.9° per column). For example, acylinder height of 5 neurons produced the following test run statisticsfor the upper left vertex class of the 2-cube SOM (see FIG. 16): 98.3%of images accurate to ±1.8°; 96.1% to ±1.3°; 81.6% to ±0.9°; 52.0% to±0.4°. We used cylinders that were 5 neurons high for the hyperalignmentruns of AQPO because these produced test runs with the lowest standarddeviation. The learning rate and number of iterations were maintained at0.001 and 300,000, respectively.

An example of a hyperaligned 3D reconstruction from the lower right handvertex partition of the N=2 case is shown in FIG. 14.

Figure Legends

FIG. 1. Flow charts for SORFA general and example case.

FIG. 2. Schematic diagram of SORFA. After entering the data set and theinitial parameters, SORFA is switched to a cylindrical topology andstarted. After training, the azimuthal angles of the data set images aredisplayed around the circumference of the cylinder. A test run isperformed to measure the consistency of the determined angles. SORFAthen is switched to an ND hypercubical array and is trained again, butthis time using an aligned set.

FIG. 3. Azimuthal angle classification of synthetic population of whitesquares. At top, is the training set to be classified by azimuthalangles. Just below, is the output cylinder cut lengthwise and laid flatfor visualization. It shows the random-pixel initial conditions, beforestarting training. During training, data set images are presented to thenetwork in random angular orientations. Output images gradually morph totraining images. Below the starting conditions is output A, aftertraining; all images along any vertical column are at the same angle butvary in size, while all images along any row are of the same size but atvarying angles. Because rotation of the output images around thecylinder is uniform, the circumferential axis may be designated as itsazimuthal axis. Training images are then classified by assigning them topositions of greatest resemblance on the output A grid (seen on left andright sides of output A). Training set images «a″ through “d″ are shownbeing mapped to closest matching grid locations. The mapping results areshown as output B. When more than one image is mapped to the same gridlocation, the average image is displayed. The azimuthal axis at thebottom shows the relative angles (just above output B label) of theimages mapped to given columns. If angular determinations are correct,images will be brought into alignment upon applying azimuthal angles(see “aligned training set”). The alignment quality is shown by thedegree of narrowness of the vertical band of images of the test run,generated when the aligned images are reprocessed through the cylinder.

FIG. 4. Azimuthal angle classification of AQPI test set. At the top arenoisy projections (SNR=0.32) of the two domains of the AQPI x-ray model,in random angular orientations. In this example, the SNR of the imagesis 0.32. These images were processed by SORFA in the same way as in FIG.3. Output B shows the cylinder portion for azimuthal angles of 22° to36°. The azimuthal axis indicates relative angles of images mapped toany given column in output B. Images were then bought into angularalignment by applying their respective azimuthal angles. The average ofthe aligned images is indistinguishable from that of the original imageswithout added noise. Finally, reprocessing the aligned images throughthe cylinder performs the test run. The degree of narrowness of thewidth of the images in the test run indicates the consistency of thealignment angles. The images representing the two domains were perfectlypartitioned to the top and the bottom of the cylinder by both output Band the test run.

FIG. 5. Effect of mapping variable numbers of test classes for N=2. Theleftmost column depicts 8 generator images used to generate 8 testclasses by adding noise and limited random rotations. Examples of imagesfor each of the eight classes are depicted in the second column (ClassesC and Dare projections of the AQP1 x-ray model). The upper right panelshows how images are mapped to diametrically opposite vertices when thedata set consists of only two principal classes. The center right panelexemplifies what occurs when the numbers of principal classes andvertices are equal: images map to four separate vertices according toclass membership. The lower right panel exemplifies what occurs whennumbers of principal classes exceeds those of vertices. The largestprincipal classes are the best competitors for the vertices, which tendto “attract’ them. Notice how the 42 image vertex class of the middlepanel does not successfully compete with the larger image classes in thelower panel, and is, consequently, not mapped to a vertex.

FIG. 6. Azimuthal angle classification of AQPO. At the top are 4 of the3,361 images comprising the C-4 symmetry imposed AQPO data (training)set. To the right is the average of all images. The circularity of theaverage image indicates that the data set is not rotationally aligned.The data set is then processed using SORFA. Output B shows the relativeazimuthal angles for all images in the training set. All the images thatmapped to a given column in output B have the same relative azimuthalangle, as indicated by the azimuthal axis. The relative azimuthal anglesare then applied to the images, producing the aligned training set, theaverage of the images of which is shown to the right. Below the set arethe results of the test run. The histogram of the test run is also shownat the very bottom. Here 82.7% of the images fall into two columns.Therefore, 82.7% of the images are precise to within 2°-4°.

FIG. 7. AQPO 2D classes. The types of classes obtained in severalstructural difference classification runs. N is the dimension used forthe N-CUBE SOM. For N=2, resulting classes of two independent runs areshown. Classes A, B, and E yielded one population of molds withcharacteristic features. Classes C and D yielded a different populationof molds with another set of characteristic features. Classes of group Fyielded molds that belonged to either population A or B.

FIG. 8. Comparison of an AQPO population A member with the extracellulardomain of the AQPI x-ray model. The top two images in the left panel,are the two views of a population A mold. The top view allowsappreciation of the depth of the four large depressions present in theconcave face of the mold. These depressions are ovular, and normal toeach other. The thick outer borders of these molds are largelyextraneous accretions of platinum, not portions of AQP0. For comparison,the extracellular domain of the x-ray model of AQP1 is at the bottom.Polypeptide loops protrude from it as four elevations. Their densitiesare also ovular, and normal to each other. The thin black lines indicatethe 25.6 Å distance between the depressions in the mold and thesimilarly spaced (24.4 Å) elevated loops in the x-ray model. The imprintof the mold, representing only half the channel, was calculated and isshown in the top right panel. Beneath it is another view of the x-raymodel, showing the entire channel. Two arrows, above both the model andthe imprint, show the elevations being compared. The distance betweenelevations in the imprint is 24.2 Å.

FIG. 9. Population B representative compared to AQPI x-ray modelcytoplasmic domain. Two views of a population B mold are depicted in theupper left panel. This mold, low pass filtered to 12.0 A, was generatedfrom a class isolated in a 2-CUBE SOM. This class mapped to a vertexopposite to that of the example of FIG. 10. The cytoplasmic side of theAQPI x-ray model is also shown at the bottom of the left panel forcomparison. Four outward pointing elevations (”fingers“) are present.Endpoints of the thin black line delineate two elevations. These are theamino termini, with a separation of 51.0 Å (indicated by black line).Elevations correlate well with rings, with a separation of 50.0 A, foundnear the mold floor (endpoints of black line). Think of the fingers asfitting into the rings. This mold's negative density would need tocorrespond to the x-ray model's positive density. FIG. 10 shows thatrings and fingers correspond to the same depth positions in the mold.The upper right panel shows the same mold, but with 7.5 A filtering.Footprints are more visible beneath the rings. The lower right panelshows another population B mold, but for N=3. The x-ray model is shownin the lower right panel. One amino terminus is circled small on thex-ray model, and one carboxylic terminus is circled large. One expectsthe carboxylic termini to leave footprint like depressions. Arrows pointto possible correlates between amino and carboxylic termini. The moldfootprint has a different orientation from that of the x-ray modelcarboxylic terminus. Rotation of this terminus would be needed to bringit into the proposed position. FIG. 11 shows that footprint dimensionsare close to those of carboxylic termini.

FIG. 10. Orientation of amino termini in a population B mold, and thepolypeptide loops in a population A mold. In the top panel, we fit thecytoplasmic half of the AQP1 x-ray model (docked into the imprint ofZampighi et al., 2003) into the mold shown in FIG. 9. The amino terminiof the x-ray model project directly through the rings of the mold andout through 4 holes in its back. In the bottom panel we juxtapose theextracellular half of the AQP1 x-ray model (docked into the imprint ofZampighi et al., 2004) next to a population A mold (shown in two viewsin FIG. 8).

FIG. 11. Analysis of footprints. The center and right columns of imagesare croppings of the rings and footprints shown in the right panels ofFIG. 9. The far right column of images contains axial views, with linesdrawn in to mark the corresponding distances. On the left side is thecytoplasmic domain of the AQP1 x-ray model. The amino and carboxylictermini are seen in the outermost perimeter. If the rings correspond tothe N termini and the footprints correspond to the C termini, then the Ctermini would need to change their orientation. One possible mechanismto account for this would be a rotation within the peptide chain, asshown.

FIG. 12. Angular deviation from linearity. The effect of the totalnumber of training cycles on the linearity of the azimuthal axis of thecylinder. We set the number of training cycles for the final run to300,000, because there is little improvement in linearity beyond thispoint.

FIG. 13. Unsymmetrized alignment and classification of AQPO. At the farleft are 6 examples of the 3,361 images that make up the unsymmetrizedAQPO data set. After alignment, a test run is performed. An N=2classification is then performed. Four 4×4 vertex classes are thengenerated from the N=2 classification and realigned separately. Anexample is shown using the upper right vertex class of the N=2classification. Output B of the second cylinder classification is thenshown. Note how the grid images resemble the final aligned average,shown at the far right. As a general rule, with proper training ofSORFA, all of output A's images tend to closely resemble both each oth rand the aligned average.

FIG. 14. Replicas generated from symmetrized and unsymmetrized alignmentand classification. In the top half of the top panel are displays of theconvex side of the replica obtained by symmetrized alignment andclassification. In the bottom half of the top panel are two views of theconcave side. On the right side of the lower panel is a mold generatedfrom the unsymmetrized AQPO data set. On the left side is thecytoplasmic domain of the AQP1 x-ray model, with the endpoints of a blueline denoting the amino termini. The endpoints of the blue line shown inthe mold on the right are two of the proposed positions for the aminotermini. The arrow shows a possible rotation in the carboxylic tail thatcould account for the orientation of the depression in the mold (alsoindicated by an arrow).

FIG. 15. Comparison of test runs using non-aligned, SORFA aligned, andMSA/MRA aligned AQPO images. The top row consists of histograms producedfrom the cylindrical grids that lie directly below. The bottom row ofimages is the averages of the images used for training the respectivetest runs. In the first column, the average is circular because theimages have not yet been aligned. The SORFA run in column two tookapproximately two hours to align. 82.74% of the images (2,781 of the3,361 images) mapped to only two cylinder columns, corresponding to analignment range of four degrees. This is manifested by an average thathas a square shape. The MSA/MRA test run in column three went throughsix rounds of alignment over a period of approximately two weeks. Thecorresponding histogram indicates that the data set has only beenpartially aligned. In the forth column are the test run results from onan independently aligned data set using MSAIMRA. The dual peaks are mostlikely a result of operator error that occurred sometime during the sixmonths period that it took to perform “alignment.” The AQPO data setused in this independently performed MSA/MRA alignment was filtered at 8A instead of 24 A (as in the other test runs), and masked more tightly.The excessive masking resulted in the images being mapped to the upperportion of the cylindrical grid.

FIG. 16. Alignment and hyperalignment. The 3,361 AQPO starting imagesare first aligned by SORFA, followed by the alignment test run (toprectangular box). Here 82.74% of the images mapped to only two cylindercolumns, corresponding to an alignment range of four degrees. Thealigned images are then used to train a 2-CUBE SOM (center) in order togenerate homogeneous classes. Four vertex classes, consisting of theimages mapped to the nine neurons at each vertex, are used here, andtheir averages are shown. Nine vertex neurons were used because thenumber of images that mapped to each of them was in the desired range of(160 180). Each vertex class is a homogeneous cluster of untiltedimages; that are needed to locate the matching tilted images, and thecorresponding azimuthal angles, needed to produce the initial 3D model.Each subset of vertex class images is then realigned using SORFA(hyperalignment), followed by a test run. This is only shown for theupper-left and lower-right vertex classes. The hyperalignment performedon the upper-left vertex class produced the following statistics: 98.32%of the vertex class images to within 3.6 degrees, 96.09% to within 2.7degrees, 81.56% to within 1.8 degrees, and 51.96% to within 0.9 degrees.The hyperalignment performed on the lower right vertex class producedthe following statistics: 78.44% to within 3.6 degrees, 68.26% to within2.7 degrees, 55.69% to within 1.8 degrees, and 32.34% to within 0.9degrees. After the new azimuthal angles are generated by hyperalignment,they are used to generate additional initial 3D models.

FIG. 16. The effect of noise on vertex class contamination. The firstrow gives a signal to noise ratio (SNR) value. The second row gives anexample of the test image with this noise level added. Rows 3 6 show theeffect of changing N on the percentage of vertex class contamination.For example, a value of 12.5% means that 12.5% of the images mapped tothe corresponding vertex is contamination from other classes.

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Although the invention has been described with reference to the aboveexamples, it will be understood that modifications and variations areencompassed within the spirit and scope of the invention. Accordingly,the invention is limited only by the following claims.

What is claimed is: 1-20. (canceled)
 21. A system for aligning andclassifying images of an object, comprising: a computer processorconfigured to: train and map a set of input 2-dimensional images to anO-dimensional cylindrical topology or geometry to obtain azimuthalangles with respect to the set of 2-dimensional images.
 22. The systemof claim 21, wherein the computer processor is further configured to:apply the azimuthal angles to the set of 2-dimensional input images toproduce aligned 2-dimensional images; and obtain a set ofclassifications based on image similarity from the aligned 2-dimensionalimages.
 23. The system of claim 22, wherein the computer processor isfurther configured to: map the set of input 2-dimensional images, or amatched set of tilted 2-dimensional images, to M-dimensional constructsbased upon the set of classifications and the azimuthal angles.
 24. Thesystem of claim 22, wherein the computer processor is configured totrain and map a class of images associated with at least oneclassification from the set of classifications to obtain a secondrefined set of azimuthal angles, thereby hyperaligning the class ofimages.
 25. The system of claim 24, wherein the computer processor isconfigured to repeat until a predetermined degree of precision isobtained for the azimuthal angles by repeating the obtaining a set ofclassifications using the hyperaligned subset of images to obtain a newset of classifications followed by training, mapping, obtainingazimuthal angles, and applying the azimuthal angles to produce alignmentthereby producing a new hyperaligned set of images.
 26. The system ofclaim 21, further comprising: receive an input of the set of2-dimensional images to the computer processor; receive an input ofmetrics to the computer processor, the metrics including a circumferenceto height ratio of the cylindrical topology, a specific circumferencebased on a desired resolution of the azimuthal angles, number oftraining cycles, and at least one parameter that impacts a degree offine-tuning of the results with each training cycle; determination ofsymmetry of the set of 2-dimensional images and an identity of theazimuthal angles based on the symmetry; and rotational alignment of theinput 2-dimensional images by applying the azimuthal angles to the input2-dimensional images.
 27. The system of claim 26, further comprisingreceiving by the computer processor of the input of the aligned2-dimensional images, wherein the receiving the input is repeated todetermine a relative precision of the azimuthal angles, wherein thedetermination is based upon how tightly constrained the azimuthal anglesare when the corresponding aligned 2-dimensional images are mapped tothe 3-dimensional cylindrical topology.
 28. The system of claim 22,wherein the processor is further configured to: receive input of thealigned images; and determine the set of classifications.
 29. The systemof claim 28 wherein at least one initial training parameter is set at avalue so that a predetermined number of input aligned 2-dimensionalimages are substantially free of a biasing effect on training.
 30. Thesystem of claim 26 wherein a 3-dimensional cylindrical topologycomprises a cylinder having a circumference and a height defining acircumference to height ratio, the circumference to height ratio beinginitially determined based on an initial small circumference, andfurther comprising scaling up the circumference while maintaining thecircumference to height ratio as constant, whereby the azimuthal anglesare obtained at higher resolution.
 31. The system of claim 21, whereinthe cylindrical topology comprises an array of prototype images thattile the cylindrical surface, wherein the array of prototype imagesinitially comprise random, or assigned, pixel values that are morphed,with each iteration of the training, in appearance toward an appearanceof one of the input 2-dimensional training images.
 32. The system ofclaim 21, further comprising performance of the training thereof basedupon a similarity between 2-dimensional prototype images and the set of2-dimensional images.
 33. The system of claim 26, wherein determinationof symmetry further comprises a review of the mapped 2-dimensionalimages to determine a degree of rotation and direction of rotation ofthe mapped 2-dimensional images with respect to the circumference of the3-dimensional cylindrical topology.
 34. The system of claim 27, whereinthe azimuthal angles are determined to be correct by checking if theazimuthal angles of the aligned images reside within a predeterminedrange of standard deviations that is acceptable for a desired precision.35. The system of claim 21, wherein the set of 2-dimensional images arealigned based on a random rotation of input images.
 36. The system ofclaim 31 wherein a most similar prototype image is selected from themorphed prototype images, and subsequently morphed based upon the mostsimilar image.
 37. The system of claim 36 wherein prototype imageslocated at a closer distance to the most similar image on a surface ofthe cylindrical topology are morphed to a greater extent than prototypeimages located at a greater distance.
 38. The system of claim 26 whereinat least one of the at least one parameter is set at a value so that apredetermined number of the input 2-dimensional images are substantiallyfree of a biasing effect on training.
 39. The system of claim 26 whereina initial learning rate is set at a value so that a predetermined numberof input 2-dimensional images are substantially free of a biasing effecton training.
 40. The system of claim 21 further comprising generating2-dimensional image data with an imaging device that images amicroscopic object.
 41. The system of claim 40 wherein the imagingdevice comprises at least one of an x-ray crystallography device or anelectron microscope.
 42. The system of claim 26 further comprising adisplay that displays the M-dimensional constructs.
 43. Acomputer-implemented method for aligning and classifying images of anobject, comprising: training and mapping a set of input 2-dimensionalimages to an O-dimensional cylindrical topology or geometry to obtainazimuthal angles with respect to the set of 2-dimensional images. 44.The method of claim 43, further comprising: obtaining a set ofclassifications based on image similarity from the set of aligned2-dimensional images.
 45. The method of claim 44, further comprising:mapping the set of input 2-dimensional images, or a matched set oftilted 2-dimensional images, to M-dimensional constructs based upon theset of classifications and the azimuthal angles.